# Finite subgroups of the unimodular group

This is related to this MO question (and others as well).

Hoping that this will not turn out to be too broad, I would like to know about the 'state of the art' of:

1) The problem of classifying finite subgroups of $\mathrm{GL}(n,\mathbb{Z})$ up to conjugacy (within $\mathrm{GL}(n,\mathbb{Z})\;$). The conjugacy classes are called "arithmetic crystal classes in dimension $n$".

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2) The problem of classifying finite subgroups of $\mathrm{GL}(n,\mathbb{Z})$ up to conjugacy (by elements of $\mathrm{GL}(n,\mathbb{Q})\;$). The conjugacy classes are called "geometric crystal classes in dimention $n$".

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Also, a couple of more vague questions:

3) For given $n$, are the above problems 1) and 2) solvable by a (sensible) algorithm? Is the difficulty of the problems more of a computational or of a conceptual nature?

It seems that, as for the classification of lattices in Euclidean space (which seems to be related), the problem presents some unexpected patterns as the dimension changes: for example, in dimension $24$ the Leech lattice appears which enjoys some uniqueness properties and an analogous is not found in other dimensions.

4) Are the classification problems 1) and 2) more "tame" as $n$ varies?

(I set "community wiki" because questions 1) and 2) may include a reference request)

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These are had questions! You probably wan to restrict to irreducible groups in 1) and 2) (in 1) that's intended to mean ireducible as a rational linear group). Question 2) is equivalent to determining the finite subgroups of ${\rm GL}(n,\mathbb{Q})$ up to equivalence. For example, Feit determned the finite irreducible subgroups of ${\rm GL}(11,\mathbb{Q})$ prior to the classification of finite simple groups. Question 2) is more amenable to attack, at least for irreducible groups. In the irreducible case, the prime factorization of $n$ is relevant, as well as the size. It's easier for prime $n$ – Geoff Robinson Mar 21 '12 at 9:19